However several people have done what you're suggesting by:
- Living on a boat (either out at sea, moored just off-shore, or at a marina). This can range from being cheaper, to being much more expensive.
- Campervan, or enclosed truck. They often park semi-legally overnight and move every few days to escape harassment. Bathrooms are problematic (gym? work?).
- Camping (i.e. just tent and woods). This often escapes harassment (since you aren't in a city) but you are limited by water, and often travelling to/from the site is time consuming if you wish to work.
- Living at work (if possible). Likely legal but can get you fired thus homeless.
- Living a hotel. Some people have claimed that they can live in a hotel for less than the cost of a mortgage + utilities + taxes + other expenses. Ditto with some cruises. YMMV.
- Living in a foreign country. If you work remotely living abroad or in the middle of nowhere can cost a LOT less than living in a big expensive city.
PS - In general being homeless is "illegal" in most US cities. You'd get harassed, maybe even arrested, just for being without a fixed address. Most cities will try to run you out of town (literally and figuratively).